Volume densities with the mean value property for harmonic functions. (English) Zbl 0818.31003

Hansen, W.; Netuka, I.

Proc. Am. Math. Soc. 123, No. 1, 135-140 (1995).

Let \(U\) be a bounded domain in \(\mathbb{R}^ d\) \((d\geq 2)\) such that \(0\in U\), and let \(\lambda\) denote Lebesgue measure on \(U\). It is shown that there exists a (strictly) positive \(C^ \infty\) function \(w\) on \(U\) such that \(h(0)= \int hw d\lambda\) for every \(h\) in \({\mathcal H}_ b (U)\), the set of bounded harmonic functions on \(U\). Further, if \(\partial U\) is of class \(C^{1+ \varepsilon}\), then it can be arranged that \(\inf w(U)>0\). The smoothness condition on \(\partial U\) cannot be omitted: the authors construct a domain \(U\) such that if \(w\) is a nonnegative measurable function on \(U\) which satisfies \(h(0)= \int hw d\lambda\) for every \(h\) in \({\mathcal H}_ b (U)\), then \(\inf w(U)= 0\). This answers a question posed by A. Cornea at the International Conference on Potential Theory in Nagoya in 1990.

Reviewer: S.J.Gardiner (Dublin)

MSC:

31B05	Harmonic, subharmonic, superharmonic functions in higher dimensions
31A05	Harmonic, subharmonic, superharmonic functions in two dimensions

Keywords:

mean value property; bounded domain; bounded harmonic functions

References:

[1]	J. Bliedtner and W. Hansen, Potential theory, Universitext, Springer-Verlag, Berlin, 1986. An analytic and probabilistic approach to balayage. · Zbl 0706.31001
[2]	Gustave Choquet and Jacques Deny, Sur quelques propriétés de moyenne caractéristiques des fonctions harmoniques et polyharmoniques, Bull. Soc. Math. France 72 (1944), 118 – 140 (French). · Zbl 0063.00852
[3]	Björn E. J. Dahlberg, Estimates of harmonic measure, Arch. Rational Mech. Anal. 65 (1977), no. 3, 275 – 288. · Zbl 0406.28009
[4]	Leopold Flatto, Functions with a mean value property, J. Math. Mech. 10 (1961), 11 – 18. · Zbl 0097.30605
[5]	Avner Friedman and Walter Littman, Functions satsifying the mean value property, Trans. Amer. Math. Soc. 102 (1962), 167 – 180. · Zbl 0103.32201
[6]	Adriano M. Garsia, A note on the mean value property, Trans. Amer. Math. Soc. 102 (1962), 181 – 186. · Zbl 0103.32202
[7]	Rudi Hirschfeld, Sur les fonctions \?-harmoniques dans un espace de mesure localement compact, C. R. Acad. Sci. Paris Sér. A-B 262 (1966), A174 – A176 (French). · Zbl 0139.06701
[8]	Emmanuel P. Smyrnélis, Mesures normales et fonctions harmoniques, Bull. Sci. Math. (2) 95 (1971), 197 – 207 (French). · Zbl 0229.31010
[9]	Kjell-Ove Widman, Inequalities for the Green function and boundary continuity of the gradient of solutions of elliptic differential equations, Math. Scand. 21 (1967), 17 – 37 (1968). · Zbl 0164.13101
[10]	Lawrence Zalcman, Mean values and differential equations, Israel J. Math. 14 (1973), 339 – 352. · Zbl 0263.35013